Show that u, v, and p are linearly independent-that is, none of the vectors is a linear combination of the other two. Three nonzero vectors u, v, and p are said to be linearly dependent if nonzero real numbers a and ẞ exist such that p = au + ẞv. Otherwise, the vectors are called linearly independent we must show that nonzero real numbers a and ẞ do not exist such that p = au + ßv. Thus, equating components in (0, -9, 17) a(1, 4, −7)+ B(2, -1, 4), we want to solve the following system. 0 -9 17 = a + 2ẞ 4α- B = -7α + 4ẞ The solution to this system is (a, b) = v, and pare = So (If an answer does not exist, enter DNE.) Thus, nonzero real numbers a and ẞ do not exist such that p = au + ẞv, so the vectors u, linearly independent.

If A = [a1, a2, a3] has as columns the three vectors a1, a2, and a3. What is y = Ax,where x is a vector with three components, x1, x2, x3? Is y in the column spaceof A?

Suppose that v = (v1, v2, ..., vn) and û = (u₁, U2, ..., Un) are a pair of n-dimensional vectors. Assume that each component of the vector is a real number, so 7 and u are both members of the set R¹. We will say that and u are "almost the same" when every component of is close to every component of ū. That is, v₁ is close to u₁, v2 is close to u2, etc (practically speaking, "close" means that their absolute difference is small). Assume that we are given the predefined predicate CloseTo(x, y) and the integer constant n. Use them to write a formal definition of the new predicate Almost The Same (7, u) which asserts that n dimensional vector is almost the same as ū. Tip: It is not legal to say i v to refer to a component of v, because is not a set. Instead, use vi to refer to the ith component of v. What set would i belong to in this case?

5. Determine if the vectors x² + x, 2x²,2x − 3, 9 in P3 are linearly independent or linearly dependent. Do these vectors span P3 ? Explain.